TIME OPTIMAL CONTROL OF FLEXIBLE SYSTEMS SUBJECT TO FRICTION
Jae-Jun Kim ∗
Rajaey Kased †
Tarunraj Singh ‡
State University of New York at Buffalo, Buffalo, NY 14260
PROBLEM FORMULATION
ABSTRACT
This paper presents a technique for the determination of timeoptimal control profiles for a rest-to-rest maneuvering multiple
spring-mass-damper system, subject to Coulomb friction. A parameterization of the control input which accounts for the friction
force, resulting in a linear analysis of the system is proposed. Optimality condition are examined for the control profile resulting
from the parameter optimization problem. The variation of the
optimal control structure as a function of final displacement is
exemplified on the friction benchmark problem.
INTRODUCTION
Mechanical systems involving relative motion are always under the influence of frictional forces. Therefore, it is important to
include the frictional effect for designing controllers when precise regulation of the system is required. However, friction phenomenon involves very complex processes such as pre-sliding,
local memory, and frictional lag.1 Because the friction force
is highly nonlinear and negatively sloped in the small velocity
region, flexible systems with vibratory motion may cause stickslip. Therefore, finding a time-optimal control for such a system
is a challenging problem. Time-optimal control of linear flexible structures can be found in numerous papers.2 Singh and
Vadali proposed a time-delay filter which shapes the step input
into bang-bang control input.3 Driessen and Sadegh formulated
a mixed integer linear programming problem to design a near
time-optimal controller including the Coulomb friction effect.4
However, finding an accurate optimal control profile with the
mixed integer programming is computationally very expensive,
and the accuracy is limited by the number of samples and convergence tolerance. In this paper, exact time-optimal control of the
single input flexible system is developed where the actuated mass
is also subject to Coulomb friction. The proposed development
illustrates that the input-shaping/time-delay filter can be used to
determine the optimal control profile for certain maneuvers. This
is possible by parameterizing an equivalent input which includes
the Coulomb friction effect. A new technique to solve for the
switch and the final times is also presented when the stiction
occurs where the input-shaping/time-delay filter cannot be used.
The optimality proof is also included for the proposed technique.
The numerical simulation is performed on a two-mass benchmark
problem, where the Coulomb friction is acting on the first mass.
The resulting controller can be applied to many applications such
as hard disk drives and flexible robot arms where the friction
force is acting on the pivot and the end effector position is to
be regulated.
∗ Graduate
Student: [email protected]
Student: [email protected]
‡ Associate Professor: [email protected]
† Graduate
x1
x2
k2 , c 2
k1 , c 1
m1
xp−1
xs
u(t)
ks−1 , cs−1
m2
ks , c s
xp
kp−1 , cp−1
kp−2 , cp−2
mp−1
ms
mp
fc
Fig. 1 General p Mass-Spring System Subject to Friction
Let us consider the minimum time control problem for a general p mass-spring-damper system shown in Figure 1. A single
control input acts on the the sth mass, which is also subject to the
Coulomb friction fc . The time optimal problem statement for a
rest-to-rest maneuver is given as
minimize
Z
tf
dt
subject to
0
ẋ(t) = Ax(t) + B u(t) − fc sign(ẋs )
x(0) = x0 , x(tf ) = xf
−U ≤ u(t) ≤ U
where A and B are defined as
0p×p
Ip×p
,
A=
−M −1 K −M −1 C
B=
(1)
0p×1
M −1 D
(2)
M , K, and C are the mass, stiffness, and damping matrices respectively. D is a control influence vector of zeros with a nonzero
value in the sth entry. fc is the Coulomb friction coefficient
which is assumed to be constant.
PARAMETERIZATION OF CONTROL
It has been previously shown by Singh and Vadali,3 that time
delay filters can be used to satisfy the boundary condition of the
rest-to-rest maneuvers. However, the systems that are analyzed
here are nonlinear systems and the concept of poles and zeros are
nonexistent. Thus the time delay filter approach cannot be used in
its current form. In order to develop a time-delay filter approach,
a linearizing net input is introduced as
unet (t) = u(t) − fc sign(ẋs )
(3)
Since the nonlinearity of the system is taken into account with
this new input, the system to be analyzed is linear with added
constraints on the parameterizations which will account for the
friction. The system equation shown in Equation (1) can be rewritten in linear form as
ẋ(t) = Ax(t) + Bunet (t)
(4)
If we parameterize the linearizing input, unet (t), instead of the
actual input, u(t), then time delay filters can be used to solve the
time optimal problem stated in Equation (1). Figure 2 shows a
American Institute of Astronautics and Aeronautics
schematic of the thought process in developing this technique for
a specific two mass spring system where the control and frictional
force are acting on the first mass. The velocity profile of the first
mass is assumed to be the first plot in Figure 2. The second plot
in
Figure 2 is the actual input to the system, assuming no singular
ag replacements
interval, to be bang bang. The new input is parameterized and
is shown as the final plot, which is the sum of actual input and
friction force. For the general multiple mass spring system shown
ẋ1 (t)
Velocity Profile
T1
T2
T3
T4
T5
tf
T kj
There are total of 2p + m constraints with n + m + 1 unknowns
(i.e. the switch and final times). Matlab’s optimization toolbox
is used to solve for the unknown switch and final times while
minimizing the final time subject to the previous constraints.
t
OPTIMALITY CONDITION
t
The time optimal control problem with the new development
in Section is written as
Z tf
minimize
dt
subject to
u(t)
Actual Input
U
0
−U
fc sign(ẋ1 )
Friction Input
+fc
t
−fc
final position of the each mass must be the same. L0 H ôpital0 s
rule is used to solve Equation (5). The next m constraints are the
velocity constraints at the time when the friction sign changes,
which are shown in Equation (6).
vj (t) = L−1 G(s)Gp (s) = 0, j = 1 . . . m
(6)
unet (t)
U + fc
U − fc
Linearizing Input
t
−U + fc
−U − fc
Fig. 2 Schematic of Linearizing Input Parameterizations for Two
Mass Spring System.
in Figure 1, we assume that the control has n switch times at the
th
`th
mass
i time instant (i = 1 . . . n) and the velocity of the s
th
is assumed to change its sign m times at the kj time instant
(j = 1 . . . m). For example, ` = [1, 3, 5] (n = 3) and k = [2, 4]
(m = 2) for the problem in Figure 2. The time delay filter which
generates the control profile when driven by a unit step input can
be parameterized as
Pn
G(s) = (U − fc ) + 2U i=1 (−1)i e−sT`i
Pm
+2fc j=1 (−1)j+1 e−sTkj + (U + fc )e−stf
G(s) must cancel out all of the poles of the system while the
control input satisfies the final boundary conditions shown in
Equation (1). One of the rigid body poles is cancelled due to
Equation (5) having a zero at s = 0. To cancel the other rigid
body pole, the time derivative of (5) must also have a zero at
s = 0. In order to cancel the flexible mode poles, Equation (5)
evaluated at sµ = σµ + iωµ should be zeros for µ = 1 . . . (p − 1).
The next constraint comes from ensuring that the final boundary
condition on position is satisfied. This can be found by using the
final value theorem. The final value theorem can be represented
as
xs (tf ) = lim G(s)Gp (s)
(5)
s→0
ẋ = Ax + B(u + (−1)j fc ),
ẋs (Tkj ) = 0
x(0) = x0 and x(tf ) = xf
−U ≤ u ≤ U
Tj−1 < t < Tj
(7)
for j = 1 . . . m and T0 = 0. It is assumed in Equation 7 that the
frictional mass starts to maneuver from rest in the positive direction with positive velocity. Equation (7) has additional interior
point constraints when the velocity of the frictional body is zero
compared to the initial problem statement in Equation (1). The
Lagrangian of this problem can be written as5
Z tf
(8)
(H − λT ẋ)dt
L = νT N +
0
where, Hamiltonian and interior point constraints are
T
j
H = 1 + λ Ax + B(u + (−1) fc ) , Tj−1 < t < Tj
N = [ẋs (Tk1 ) ẋs (Tk2 ) . . . ẋs (Tkm )]T = 0
(9)
for j = 1 . . . m and the Lagrangian multipliers are defined as
≥ 0 t = T kj
j = 1...m
(10)
νj
= 0 elsewhere
Since the Hamiltonian is not explicitly a function of time, it is
equal to zero for all time t. As in the system shown in Equation
(1), the switching function for a new problem is also given as
B T λ(t). We also know that for optimality, the switching function
must cross zero at the switching times such that
B T λ(T`i ) = 0,
i = 1...n
(11)
The co-state equation from the optimality condition becomes
λ̇ = −
∂H
= −AT λ,
∂x
for all time except t = Tkj
(12)
for j = 1 . . . m. when t = Tkj , the co-states should satisfy the
following equations.
∂N
where Gp (s) is the transfer function of the sth mass position output derived from Equation (4). For rest to rest maneuvers, the
λ(Tk−j ) = λ(Tk+j ) + νj ∂x(Tkj
∂N
j
H(Tk−j ) = H(Tk+j ) − νj ∂Tkj
American Institute of Astronautics and Aeronautics
j
)
,
j = 1...m
(13)
Because N is not explicitly a function of time, the Hamiltonian
should be continuous such that H(Tk−j ) = H(Tk+j ). Therefore,
jump discontinuity in the co-state has to be chosen to satisfy the
continuous Hamiltonian requirements, which can be written in
the form
∂Nj
(14)
νj
= γj B T λ(Tk−j )
∂x(Tkj )
Once νj ’s are determined, we can find an expression for λ(T`i ),
in terms of the initial co-states, λ(0), from Equation (12) and
(13). Then we can build Equation (11) in terms of λ(0). As an
example, assume that the first time that the velocity goes to zero
is Tk1 , then the co-states at Tk−1 are given by equation (15).
λ(Tk−1 ) = e−A
T
Tk−
1
(15)
λ(0)
The co-states at t = Tk+1 are given as
T
Tk−
1
λ(0)
(16)
where, J1 is a 2p × 2p matrix where (p + s)t h row is −γj B T and
zeros for the rest, and I is a 2p × 2p identity matrix. Then the costates can be integrated until the next switch with this new initial
conditions. This procedure is repeated until the n × 2p matrix M
is formed for n switch times such that

 T
B λ(T`1 )
 B T λ(T`2 ) 
 = Mλ(0) = 0

(17)

 ···
B T λ(T`n )
λ(0) is found by the null space of M which satisfies the Hamiltonian requirements such that H(t = 0) = 0. After calculating
λ(0) from Equation (17), the co-states can then be integrated forward and the resulting switching function must cross zero at the
switching time to ensure optimality.
PSfrag replacements
NUMERICAL EXAMPLE
|u| ≤ 1
x2
k=1
m1 = 1
m2 = 2
fc = 0.4
Fig. 3 Two Mass Harmonic System Subject to Coulomb Friction
The example problem considered is illustrated in Figure 3. The
control input and Coulomb friction forces are acting on the first
mass. The time optimal control problem can be stated as
Z
tf
subject to
 
0
0
1 0
0 0
0

0
0
1




x(t) +   u − fc sign(x3 )
ẋ(t) = 
−1
1
0 0
1
0.5 −0.5 0 0
0
(18)
T
T
x(0) = 0 0 0 0 , x(tf ) = 1 1 0 0 , −1 ≤ u ≤ 1
minimize

dt
0
subject to the switch time constraints,
0 ≤ T 1 ≤ T2 ≤ T3 ≤ T4 ≤ T5 ≤ tf
(20)
rigid body pole cancellation constraint,
2T1 − 2fc T2 − 2T3 + 2fc T4 + 2T5 − (1 + fc )tf = 0
(21)
flexible body pole cancellation constraints,
λ(Tk+1 ) = (I + J1 )λ(Tk−1 ) = (I + J1 )λe−A
x1
With the knowledge that the time optimal control profile of the
linear undamped two mass spring system has 3 switches and 2
zero velocity crossings, we assume n = 3, m = 2, k = [2 4],
` = [1
√ 3 5]. The flexible poles are given by s = ±ωi where
ω = 1.5. The net control input is parameterized as shown in
Figure 2. Then the parameter optimization problem can be stated
as:
minimize tf
(19)
(1 − fc ) − 2U cos(ωT1 ) + 2fc cos(ωT2 ) + 2 cos(ωT3 )
−2fc cos(ωT4 ) − 2 cos(ωT5 ) + (1 + fc ) cos(ωtf ) = 0
(22)
−2 sin(ωT1 ) + 2fc sin(ωT2 ) + 2 sin(ωT3 )
−2fc sin(ωT4 ) − 2 sin(ωT5 ) + (1 + fc ) sin(ωtf ) = 0
(23)
final displacement constraint,
1
(−2T12 + 2fc T22 + 2T32 − 2fc T42 − 2T52 + (1 + fc )t2f ) = 1 (24)
4
and first mass velocity constraints,
h
1
m2
(1 − fc )
sin ωT2 + T2
m1 + m 2
m1 ω
i
m2
−2
sin ω(T21 ) + (T21 ) = 0
m1 ω
h
1
(1 − fc ) mm12ω sin ωT4 + T4
m1 + m 2
m2
− 2
sin ω(T41 ) + (T41 )
m1 ω
m2
sin ω(T42 ) + (T42 )
+ 2fc
m ω
i
m2 1
sin ω(T43 ) + (T43 ) = 0
+ 2
m1 ω
(25)
(26)
where Tab stands for Ta − Tb in Equations (25) and (26). The
constrained nonlinear optimizer of MatLab was used to solve
this problem. The time-delay filter is found to be
G(s)
= 0.6 − 2e−1.8164s + 0.8e−2.1177s + 2e−3.1498s
−0.8e−3.5918s − 2e−4.4047s + 1.4e−5.2298s
(27)
Co-states and switching curve can be computed with the resulting
control profile to determine the optimality. The Hamiltonian of
the problem is

 1 + λT Ax + B(u − fc ) 0 < t < T2
1 + λT Ax + B(u + fc ) T2 < t < T4
H=
(28)

1 + λT Ax + B(u − fc ) T4 < t < tf
and the interior point constraints are
N1 = ẋ1 (T2 ) = 0
American Institute of Astronautics and Aeronautics
and
N2 = ẋ1 (T4 ) = 0.
(29)
ẋ1
B T λ, H, unet , u
1.5
The Hamiltonian should be continuous such that H(T2+ ) =
H(T2− ) and H(T4+ ) = H(T4− ). The switching function is
1
BT λ
B T λ = λ3 and there is a discontinuity only in λ3 . The con0.5
tinuous Hamiltonian requirement yields
PSfrag replacements
0
H
1 + [λ1 (T2 ) λ2 (T2 ) λ3 (T2− ) λ4 (T2 )] Ax(T2 ) + B(−1 − fc ) −0.5
= 1 + [λ1 (T2 ) λ2 (T2 ) λ3 (T2+ ) λ4 (T2 )] Ax(T2 ) + B(−1 + fc )
−1
u
(30)
unet
1 + [λ1 (T4 ) λ2 (T4 ) λ3 (T4− ) λ4 (T4 )] Ax(T4 ) + B(1 + fc ) −1.5
0
1
2
3
4
5
6
= 1 + [λ1 (T4 ) λ2 (T4 ) λ3 (T4+ ) λ4 (T4 )] Ax(T4 ) + B(1 − fc )
Time (sec)
(31)
Assuming λ3 (T2+ ) = λ3 (T2− ) + γ1 λ3 (T2− ) and λ3 (T4+ ) =
Fig. 4 Control Input, Switching Function, and Hamiltonian
λ3 (T4− ) + γ2 λ3 (T4− ), γ1 and γ2 are found to be
the velocity of the first mass should cross zero before the new
−2fc
control switch activates. This implies that now T3 is the time
γ1 =
−1 + fc − k x1 (T2 ) − x2 (T2 )
when
the velocity is zero and T4 is the time when the control
(32)
2fc
switches.
At T3+ , the velocity becomes positive and therefore
γ2 =
1 − fc − k x1 (T4 ) − x2 (T4 )
friction force becomes −fc . However, the linearizing input with
the new friction value drives the first mass to the negative direcNow the co-states at the switch times are found in terms of λ(0)
tion if the net input is less than zero. Once the first mass crosses
such that
T
the zero velocity, the net input changes its sign again because of
λ(T1 ) = e−A T1 λ(0)
the change in friction value to +fc . This “chattering” is equivaT
λ(T2− ) = e−A (T2 −T1 ) λ(T1 )
2
λ(T2+ ) = (I + J1 )λ(T2− )
T
1.5
(33)
λ(T3 ) = e−A (T3 −T2 ) λ(T2+ )
T
1
λ(T4− ) = e−A (T4 −T3 ) λ(T3 )
+
−
λ(T4 ) = (I + J2 )λ(T4 )
0.5
T
λ(T5 ) = e−A (T5 −T4 ) λ(T4+ )
0
where, J1 and J2

0 0
 0 0
J1 = 
 0 0
0 0
is defined as

0 0
0 0 
 and
γ1 0 
0 0
PSfragreplacements −0.5
0 0
−1
0
1
2
3
4
5
6
7
8
0 0 
Time (sec)

γ2 0 
Fig. 5 Velocity Profile After Transition Displacement
0 0
(34)
lent to the stiction caused by the hard nonlinearities in the friction
M is now given to satisfy Mλ(0) = 0 when t = [T1 , T3 , T5 ].
model and it will continue until the sum of all the forces to the
The initial co-states are found to satisfy Mλ(0) = 0 and H(t =
frictional body overcomes the friction force. The velocity of the
0) = 0 such that
first mass for this chattering region becomes zero, which will be
the additional constraints to the problem. For the example probNull(M)
λ(0) = −
(35)
lem, the predicted velocity profile of the first mass is plotted in
[Null(M)]T B(1 − fc )
Figure 5 showing that the velocity of the first mass is zero until
where, Null(M) is the null space of M. The linearizing input, as
the new switch actuates. With the velocity profile in Figure 5, the
well as the switching curve is shown in figure 4 for the example
new optimal control problem with state equality constraints can
problem. The actual input is found by subtracting the friction
be formulated as5
force from the net input.
Z tf
dt
subject to
minimize

0
 0
J2 = 
 0
0
0
0
0
0
VARIATION OF CONTROL STRUCTURE
In the previous example, the velocity of the first mass is zero
at T2 and T4 . As the final position increases for the previous example, the time gap between T3 and T4 becomes small until they
coincide. The control profile for this transition indicate that the
velocity crossing coincides with the control switch which will result in a four switch control profile. The transition displacement
for this example is d ≈ 3.4. After the transition displacement, going back to original profile does not provide a feasible solution.
Also, keeping the four switch transition profile after the transition
displacement, creates an over-constrained problem. Therefore,
0
ẋ = Ax + B u − fc , 0 < t < T2
ẋ = Ax + B u + fc , T2 < t < T3
ẋ = Ax + Bu, T3 < t < T4
ẋ = Ax + B u − fc , T4 < t < tf
k
(x2 − x1 ) + u = 0,
m1
ẋ1 (T2 ) = ẋ1 (T3 ) = 0
x(0) = x0 and x(tf ) = xf
−1 ≤ u ≤ 1
ẍ1 =
American Institute of Astronautics and Aeronautics
T 3 < t < T4
(36)
Because the friction force is zero for the zero velocity, the control
for T3 < t < T4 should satisfy the equality constraint in Equation
36 such that
u=
k
(x1 − x2 ),
m1
T3 < t < T 4
(37)
Therefore, the control profile can be parameterized from the predicted velocity profile including the state constraint effect during
the chattering interval. The time-delay filter cannot be used for
this profile because of the feedback control effect in the chattering
interval which alters the system equation. However, the control
profile can be used to integrate the system forward in time using
the given initial guesses of the switch times to obtain the states at
the switch and final times.
Z T1
eA(t−τ ) B(1 − fc )dτ
x(T1 ) = eAT1 x0 +
0
Z T2
A(T2 −T1 )
x(T2 ) = e
x(T1 ) +
eA(t−τ ) B(−1 − fc )dτ
T
1
Z T3
x(T3 ) = eA(T3 −T2 ) x(T2 ) +
eA(t−τ ) B(−1 + fc )dτ
x(T4 ) = eA
x(T5 ) = e
c
(T4 −T3 )
A(T5 −T4 )
T2
x(T3 )
x(T4 ) +
x(tf ) = eA(T6 −T5 ) x(T5 ) +
Z
(38)
T5
e
Z TT46
T5
A(t−τ )
B(1 − fc )(τ )dτ
eA(t−τ ) B(−1 − fc )(τ )dτ
where, Ac is the closed loop system equation in the interval at
t ∈ [T3 , T4 ] using Equation 37. For the computational accuracy
and convenience, it is very useful to use the following property
(van Loan identity) to compute the states at the switch times.


Z t
A(t−τ
)
At
A B
e
Bdτ 
e
(39)
t =
exp
0
0 0
0
1
Now the problem can be solved for switch times numerically by
minimizing the final time T6 with the constraints
ẋ1 (T2 ) = 0, ẋ1 (T3 ) = 0
x(tf ) = [1 1 0 0]T
(40)
The resulting control profile is used to verify the optimality of
the control input by inspecting the optimality conditions. The
lagrangian for this problem can be formulated as5
Z tf
(H − λT ẋ)dt
L = ψN1 + πN2 +
(41)
t0
where Hamiltonian and constraints are
H = 1 + λT Ax + B(u − fc ), 0 < t < T2
H = 1 + λT Ax + B(u + fc ) , T2 < t < T3
H = 1 + λT (Ax + Bu) + µC, T3 < t < T4
H = 1 + λT Ax + B(u − fc ) , T4 < t < tf
N1 = ẋ1 (T2 ) = 0
N2 = ẋ1 (T3 ) = 0
C = ẍ1 (t) = k(x2 − x1 ) + u = 0
(42)
and the Lagrangian multipliers are defined as
≥0
≥ 0 t = T2
and π
ψ
=0
= 0 elsewhere
≥ 0 t ∈ [T3 , T4 ]
µ
= 0 elsewhere
t = T3
(43)
elsewhere
(44)
The input u is considered to be within the saturation limit for
T3 < t < T4 because of the stiction assumption with u = −1.
Then, µ can be found from the following relationship.
∂H
= 0 = BT λ + µ
∂u
or
µ = −B T λ,
T3 < t < T 4
(45)
Therefore, the co-state equation is found from Equation 45 and
the necessary optimality condition.
 ∂C T
∂H  −AT +
λ T3 < t < T 4
B
=
(46)
λ̇ = −
∂x

∂x
T
−A λ elsewhere (t 6= T2 , t 6= T3 )
At the interior points, t = T2 and t = T3 , the co-states becomes
discontinuous and are defined by the following equations.
∂N1
∂N2
λ(T2− ) = λ(T2+ ) + ψ ∂x(T
, λ(T3− ) = λ(T3+ ) + π ∂x(T
2)
3)
−
+
−
+
∂N1
∂N2
H(T2 ) = H(T2 ) − ψ ∂T2 , H(T3 ) = H(T3 ) − π ∂T3
(47)
Because the interior point constraints are not explicitly functions of time, the Hamiltonian is continuous such that H(T2− ) =
H(T2+ ) and H(T3− ) = H(T3+ ). Therefore, ψ and π in Equation
47 are chosen to satisfy the continuous Hamiltonian requirement
which will yield the following equations
γ1 =
−2fc
,
−1 + fc − k x1 (T2 ) − x2 (T2 )
λ3 (T3− ) = 0
(48)
where, γ1 satisfies λ3 (T2+ ) = λ3 (T2− )+γ1 λ3 (T2− ). Co-states can
be integrated forward in time with the initial λ(0) using Equation
46 and 48.
T
λ(T1 ) = e−A T1 λ(0)
T
λ(T2− ) = e−A (T2 −T1 ) λ(T1 )
λ(T2+ ) = (I + J1 )λ(T2− )
T
λ(T3− ) = e−A (T3 −T2 ) λ(T2+ )
+
λ(T3 ) = (I + J2 )λ(T3− )
T
∂C
λ(T4 ) = e(−A + ∂x )(T4 −T3 ) λ(T3+ )
T
λ(T5 ) = e−A (T5 −T4 ) λ(T4 )
Where, I is an 4 × 4 Identity Matrix and



0 0 0 0
0
 0 0 0 0 
 0


J1 = 
 0 0 γ 1 0  , J2 =  0
0 0 0 0
0
0
0
0
0

0 0
0 0 

∞ 0 
0 0
(49)
(50)
Because λ(T3− ) should be zero, infinite value has to be multiplied
to λ(T3− ) in J2 to have finite jump discontinuities in the co-states.
For numerical computations, a large number is used in J2 instead
of ∞. Since the switching curve should cross zero at the actual
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switch times, T1 , T4 , and T5 , M can be determined such that
Mλ(0) = 0. Then, λ(0) is selected to satisfy Hamiltonian requirement, H(t = 0) = 0, such that
λ(0) = −
Null(P )
[Null(P )]T B(1 − fc )
1
B T λ, H, unet , u
It is shown that the linearizing input approach provides the
time optimal control solution of the multiple mass spring damper
system under the assumption of correct parametrization of the
control input. The frictional benchmark problem results in a timeoptimal three-switch bang-bang control profile. The linearizing
input parametrization technique can also extended to systems
where one or more mass is under frictional forces. This will require a correct knowledge of velocity profiles of all the masses
subject to the friction.
BT λ
References
1 Pierre
0.5
H
0
T1 T2
T4
T3
T5
−0.5
u
−1
unet
−1.5
−2
CONCLUSION
(51)
In Figure 6, the linearizing and actual control input along with the
switching curve is plotted. The actual input during the stiction violates the control input limits, however, it is possible to maintain
the control to be −1 and let the available friction force the system to stick. Then the actual control profile becomes three-switch
bang-bang. The resulting control input satisfies the necessary optimality conditions as shown in Figure 6.
placements
that the control switches are smooth curves for final displacement changes. The dotted line in Figure 7 denotes the velocity
switches.
0
1
2
3
4
5
6
7
8
9
Dupon Brian Armstrong-Helouvry and Carlos Canudas De Wit. Survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica, 30(7):1083–1138, 1994.
2 S. L. Scrivener and R. C. Thompson. Survey of time-optimal attitude maneuvers. Journal of Guidance, Control, and Dynamics, 17(2):225–233, March-April
1994.
3 T. Singh and S. R. Vadali. Robust time-optimal control: Frequency domain
approach. Journal of Guidance, Control, and Dynamics, 17(2):346–353, MarchApril 1994.
4 Brian J. Driessen and Nader Sadegh. Minimum-time control of systems with
coulomb friction: near global optima via mixed integer linear programming. Optimal Control Applications and Methods, 22(2):51–62, July 2001.
5 Jr. Arthur E. Bryson and Yu-Chi Ho. Applied Optimal Control, chapter 3.
Hemisphere Publishing Corporation, 1975.
Time (sec)
Fig. 6 Linearizing Control Input, Switching Function, and Hamiltonian
As the final displacement is further increased, the profile of
the control shows further transitions. The control profile can be
solved similarly as in the previous development. The summary
of the control input transition profile for the example problem is
shown in Figure 7. The actual control switch times are plotted
15
10
placements
T ’s
5
0
I
0
2
II
4
III
6
8
10
IV
12
14
ẋ1
unet
u
Region I
Region II
Region III
Region IV
Fig. 7 Switch Time vs. Displacement with Velocity and Control
Input Profiles
with the solid line as a function of final displacement showing
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